DEFORMATION OF 2-STEP NILMANIFOLDS WITH ABELIAN … · a(2n + 1)-dimensional Heisenberg group, then...

J. London Math. Soc. (2) 73 (2006) 173–193 C�2006 London Mathematical Societydoi:10.1112/S0024610705022519

DEFORMATION OF 2-STEP NILMANIFOLDS WITH ABELIANCOMPLEX STRUCTURES

C. MACLAUGHLIN, H. PEDERSEN, Y. S. POON and S. SALAMON

Abstract

We develop deformation theory for abelian invariant complex structures on a nilmanifold, andprove that in this case the invariance property is preserved by the Kuranishi process. A purelyalgebraic condition characterizes the deformations leading again to abelian structures, and weprove that such deformations are unobstructed. Various examples illustrate the resulting theory,and the behavior possible in three complex dimensions.

1. Introduction

In this paper, we study complex structures associated to compact quotients ofnilpotent Lie groups. These manifolds are called nilmanifolds, and an investigationof the special class of Kodaira manifolds was completed in [13]. The present paperopens this discussion to a wider class of nilmanifolds.

A left-invariant complex structure on a Lie group is said to be abelian if thecomplex space of (1,0)-vectors is an abelian algebra with respect to Lie bracket.This concept was first considered in [5], and plays a fundamental role in the studyof complex structures on solvable Lie groups [3, 4], turning out to be especiallyimportant in hypercomplex and HKT geometry [2, 9, 10, 14]. Because the definitiononly makes sense in the algebraic setting, it is of particular interest to know to whatextent a study of invariant complex structures on nilmanifolds captures the generalsituation.

There is a total of six 6-dimensional 2-step nilpotent groups admitting abeliancomplex structures [8, 21]. IfRn denotes an n-dimensional abelian group andH2n+1

a (2n + 1)-dimensional Heisenberg group, then the 2-step nilpotent groups withabelian complex structures are R6, H5 × R1, H3 × R3, H3 × H3, the Iwasawagroup W6 and one additional group which we denote by P6. These groups are the6-dimensional instances of the respective series: R2n, H2n+1 × R2m−1, H2n+1 ×H2m+1, W4N+2 and P4N+2. For example, W4N+2 is the real group underlying ageneralized complex Heisenberg group.

The compact quotients ofR2n are complex tori, and their deformation and moduliare well studied [6]. A detailed account of the moduli space of complex structuresof a special compact quotient of H2n+1×R was recently given in [13], and in [15] asomewhat different method is directed towards the study of W6. The present paperhelps to unite these two approaches, particularly via the examples in § 6.

This paper presents a general approach to computing the deformations of 2-stepnilmanifolds with abelian complex structures. To analyze data on deformation

Received 10 June 2004.

2000 Mathematics Subject Classification 32G05; 53C15, 53C56, 57S25, 17B30.

CM and YSP are partially supported by NSF DMS-0204002. HP and SS are partially supportedby EC contract HPRN-CT-2000-00101.

174 C. MACLAUGHLIN, H. PEDERSEN, Y. S. POON AND S. SALAMON

theory, our first step is to identify the Dolbeault cohomology of a 2-step nilmani-fold with abelian complex structure with the appropriate Lie algebra cohomology(Theorem 1). General results of this nature were proved in [7], though we shallneed our own explicit description of this identification. The second step is to extendthis identification to the determination of harmonic representatives for Dolbeaultcohomology with coefficients in the tangent sheaf (Theorem 3).

We derive the following first main result of this paper in § 4.3.

Theorem. Let G be a 2-step nilpotent Lie group with co-compact subgroup Γ.Then any abelian invariant complex structure on X = Γ\G has a locally completefamily of deformations consisting entirely of invariant complex structures.

This is proved by showing that an application of Kuranishi’s method does nottake one outside the subspace of invariant tensors. The theorem implies that anydeformation of an abelian invariant complex structure is necessarily equivalent toan invariant one, at least if the deformation is sufficiently small.

Given this result, it makes sense to ask under what conditions the deformedinvariant structures remain abelian. Indeed, our techniques enable us to prove thenext result, that deformations preserving the abelian property are always unob-structed and faithfully represented at the infinitesimal level.

Theorem. On a 2-step nilmanifold X with an abelian complex structure, avector in the virtual parameter space H1(X,ΘX) is integrable to a 1-parameterfamily of abelian complex structures if and only if it lies in a linear subspace definingthe abelian condition infinitesimally.

The ‘only if’ part is obvious, but the force of this result is the backwards impli-cation. Once we convert to Lie algebra cohomology, it reduces the constraintsof abelian deformations to purely algebraic equations that we introduce in § 5and collectively call ‘Condition A’. Using this, one may carry out an effectivecomputation in terms of structural constants of the nilpotent groups in question.

Further analysis of abelian deformations yields a characterization of the Kodairamanifolds (defined in § 2.1) as those corresponding to a Lie algebra with a center ofcomplex dimension 1 and for which all infinitesimal parameters are integrable andlead to abelian deformations. The precise statement is Theorem 6 in § 5.1.

In the final section, we compute the relevant cohomology dimensions for a numberof nil 6-manifolds, each equipped with a natural abelian complex structure. In sodoing, we are able to compare the techniques of this article with those of [21],but we emphasize a complication that arises from a choice of complex structurewith added symmetry. The first theorem above allows one to dispense with theKuranishi method in the explicit construction of parameter spaces, and to replaceit with a more direct calculation involving invariant differential forms. This we doin Example 8 in § 6.1, after having first illustrated the power of the second theoremabove in estimating the dimension of a potential moduli space.

2. Abelian complex structures

Suppose that a Lie algebra g admits an endomorphism J such that

J ◦ J = −1 and [JA, JB] = [A,B] (1)

DEFORMATION OF 2-STEP NILMANIFOLDS 175

for all A,B in g. It can be extended by left-translation to an endomorphism of theentire tangent bundle of G. Then J defines an invariant almost complex structureon the group G which is integrable, since (1) implies the vanishing of the Nijenhuistensor. A complex structure satisfying (1) is called abelian, and the identity alsoimplies that the center c is J-invariant.

Now assume that the Lie algebra is 2-step nilpotent. In particular, the first derivedideal [g, g] is contained in the center. Taking the quotient of the algebra g withrespect to the center, we obtain an abelian algebra t. When the complex structureis abelian, it induces a complex structure on t. The identities

g(1,0) = t(1,0) ⊕ c(1,0) and g(0,1) = t(0,1) ⊕ c(0,1)

concerning type (1,0)- and (0,1)-vectors are therefore valid at the level of vectorspaces.

Let {Xi, JXi : 1 � i � n} be a real basis for t and {Zα, JZα : n+1 � α � n+m}a real basis for c. A basis of (1, 0) vectors for the complex tangent bundle of G iscomposed of the elements

Tj = 12 (Xj − iJXj) and Wα = 1

2 (Zα − iJZα). (2)

The complex structural constants Eαkj and Fα

kj are defined by

[T k, Tj ] =∑

α

EαkjWα +

∑

α

FαkjWα, (3)

and satisfy

Fαkj = −Eα

jk.

We continue to use roman indices in the range 1, . . . , n and Greek indices forn + 1, . . . , n + m. Let ωj be the left-invariant (1,0)-forms dual to the vectors Tj

for 1 � j � n and annihilating the Wα. They span the space t∗(1,0). Similarly, thereare left-invariant (1,0)-forms ωα dual to Wα, annihilating the Tj . The dual form ofthe structural equation (3) is

dωα =∑

i,j

Eαjiω

i ∧ ωj .

The forms ωi are all exact, and ∂ωα = 0. Thus, we have the following lemma.

Lemma 1. The forms ω1, . . . , ωn, ωn+1, . . . , ωn+m are all ∂-closed.

Now suppose that there exists a discrete subgroup Γ of G such that the leftquotient space Γ\G is compact. The resulting quotient is called a nilmanifold. Sincethe complex structure J is left-invariant, it descends to a complex structure onX = Γ\G. Such a discrete subgroup always exists if there is a basis such that thereal structural constants are rational [18].

Later in this paper, we shall study the deformation theory on such compactcomplex manifolds. At an appropriate juncture (in § 4.1), we shall find it convenientto introduce an invariant Hermitian metric on X . We shall choose such a metric sothat {Xj, JXj , Zα, JZα} forms a Hermitian frame. First we describe some simpleexamples of nilmanifolds and complex structures.


2.1. Kodaira manifolds and other examples

Our first example of a compact nilmanifold with an abelian complex structureis a Kodaira manifold, a generalization of a Kodaira surface. We proceed to listalgebraic constructions of this and similar examples.

Example 1. On the vector space R2n+2 with basis {Xj , Yj , Z,A}, define a Lie

algebra by setting

[Xj, Yj ] = −[Yj , Xj ] = Z, 1 � j � n,

and declaring all other brackets to be zero. This turns the vector space into thedirect sum g = h2n+1 ⊕ t1 of the Heisenberg algebra and the 1-dimensional algebra.

We define an almost complex structure J on the Lie algebra g by means of theequations

JXj = Yj , JZ = A, (4)

so that the equations JYj = −Xj and JA = −Z are also understood. The endomor-phism J defines an abelian complex structure on g, and therefore on the manifoldH2n+1×R and a compact quotient thereof. In the notation (2), its complex structureequation is

[T j , Tj ] = − 12 i(W +W ). (5)

The moduli problem of the compact quotient of such complex manifolds was studiedextensively in [13].

We next describe a more general extension of the Heisenberg group, and then aproduct example.

Example 2. Let H2n+1 × R2m+1 be the product of a real Heisenberg groupand an abelian Lie group with dimensions as specified. Let {Xj, Yj , Z} be a basisfor h2n+1, and let {Z0, Z2k−1, Z2k} (with 1 � k � m) be a basis for R2m+1. Thenon-zero Lie brackets are determined by the single set of equations

[Xj , Yj ] = Z.

An abelian complex structure is defined by setting

JXj = Yj , JZ = Z0, JZ2k−1 = Z2k, (6)

in analogy to (4).

Example 3. The product H2n+1 ×H2m+1 is a 2-step nilpotent group with real2-dimensional center. Let {Xj , Yj , Z1, Xk, Yk, Z2} (with 1 � j � n, 1 � k � m) bea basis for its Lie algebra h2n+1 ⊕ h2m+1. The non-zero Lie brackets are

[Xj , Yj ] = Z1 and [Xk, Yk] = Z2.

Define an almost complex structure J on this space by

JXj = Yj , JXk = Yk, JZ1 = Z2; (7)

once again this defines an abelian complex structure.

We describe the remaining classes of examples in 6 dimensions for simplicity.


Example 4. The group structure of W6 underlies that of the complex Heisen-berg group. On the algebra level, the Lie brackets of W6 are

[X1, X3] = − 12Z1, [X1, X4] = − 1

2Z2, [X2, X3] = − 12Z2, [X2, X4] = 1

2Z1. (8)

An abelian complex structure is defined by

JX1 = X2, JX3 = −X4, JZ1 = −Z2, (9)

and this is denoted J1 in [15]. Beware that J is not the standard bi-invariantcomplex structure J0 that makes W6 a complex Lie group; indeed

J0[A,B] = [J0A,B], A,B ∈ g,

and so J0 is definitely not abelian. Nonetheless, both J0 and J1 induce the sameorientation on W6.

Example 5. The Lie brackets for P6 are given by

[X1, X2] = − 12Z1, [X1, X4] = − 1

2Z2, [X2, X3] = − 12Z2,

and correspond to a degeneration of (8). An abelian complex structure J on P6 isdefined by

JX1 = X2, JX3 = −X4, JZ1 = −Z2. (10)

The associated Hermitian manifold Γ\P6 was studied in [1, § 5], where it was shownthat J is only one of a finite number of complex structures compatible with a fixedRiemannian metric.

3. Cohomology theory

In order to perform deformation theory on the compact complex nilmanifold X ,we need to calculate cohomology with coefficients in the holomorphic tangent sheaf.We achieve this by identifying Dolbeault and Lie algebra cohomology, in the spiritof [20].

3.1. Lie algebra cohomology

With respect to a complex structure J , the complexified Lie algebra has a typedecomposition. We may write

gC = g1,0 ⊕ g0,1, tC = t1,0 ⊕ t0,1, cC = c1,0 ⊕ c0,1.

These are all spaces of left-invariant vectors on G. The definitions are extended toinvariant (p, q)-forms in the standard way. For instance,

∧kg∗(0,1)C

= g∗(0,k) is thespace of G-invariant (0, k)-forms.

Motivated by the property of the Chern connection on holomorphic tangentbundles [12], we define a linear operator ∂ on (0, 1)-vectors as follows. For any(1, 0)-vector V and (0, 1)-vector U , set

∂UV := [U, V ]1,0.

We obtain a linear map

∂ : g1,0 → g∗(0,1) ⊗ g1,0.


In view of (3),

∂T kTj = [T k, Tj]1,0 =

∑

α

EαkjWα, (11)

whence

∂Tj =∑

k,α

Eαkjω

k ⊗Wα and ∂Wα = 0.

We extend this definition to a linear map on g∗(0,k) ⊗ g1,0 by setting

∂(ω ⊗ V ) = ∂ω ⊗ V + (−1)kω ∧ ∂V,where ω ∈ g∗(0,k) and V ∈ g1,0. For instance, any element µ in g∗(0,1) ⊗ g1,0 can bewritten as

µ =∑

i,j

µijω

j ⊗ Ti +∑

i,α

µiαω

α ⊗ Ti +∑

j,β

µβj ω

j ⊗Wβ +∑

α,β

µβαω

α ⊗Wβ . (12)

By Lemma 1 and (11),

−∂µ =∑

i,j

µijω

j ∧ ∂Ti +∑

i,α

µiαω

α ∧ ∂Ti

=∑

i,j,k,β

µijE

βkiω

j ∧ ωk ⊗Wβ +∑

i,k,α,β

µiαE

βkiω

α ∧ ωk ⊗Wβ .

This calculation gives us a necessary and sufficient condition for µ to be ∂-closed,which we now record as our next lemma.

Lemma 2. Suppose that an element µ in g∗(0,1) ⊗ g1,0 is given by (12). Then∂µ = 0 if and only if

∑i(µ

ijE

αki − µi

kEαji) = 0 and

∑i µ

iαE

βji = 0,

for each j, k, α, β.

We have a sequence

0 → g1,0 → g∗(0,1) ⊗ g1,0 → · · · → g∗(0,k−1) ⊗ g1,0 ∂k−1−−−→ g∗(0,k) ⊗ g1,0 ∂k−→ · · · .The next result comes as no surprise, reflecting as it does the fact that our∂-operators are the natural ones induced on invariant differential forms.

Lemma 3. The above sequence is a complex, i.e. ∂k ◦ ∂k−1 = 0 for all k � 1.

Proof. It suffices to verify the lemma for k = 1. Let {ωp}, {eq} be dual bases ofg∗(0,1) and g0,1, where the indices p, q run over the entire range 1, . . . , n + m. Bydefinition,

∂V =∑

p

ωp ⊗ [ep, V ]1,0.

Applying ∂ again,

∂2V =

∑

p

∂ωp ⊗ [ep, V ]1,0 −∑

p,q

(ωp ∧ ωq) ⊗ [eq, [ep, V ]1,0]1,0.


Since [ep, eq]1,0 = 0, we can delete the penultimate projection 1,0 above. The Jacobiidentity

[ep, [eq, V ]] − [eq, [ep, V ]] = [[ep, eq], V ]

implies that∑

p,q

ωp ∧ ωq ⊗ [eq, [ep, V ]]1,0 = −12

∑

p,q

ωp ∧ ωq ⊗ [[ep, eq], V ]1,0.

If σ is a (1, 0)-form, we can now contract σ with ∂2V to obtain the following form

of type (0, 2):

σ(∂2V ) =

∑

p

σ([ep, V ])∂ωp +12

∑

p,q

σ([[ep, eq], V ])(ωp ∧ ωq)

= −2∑

p

dσ(ep, V )∂ωp −∑

p,q

dσ([ep, eq], V )(ωp ∧ ωq).

For this to vanish for all V and σ, we need to show that

2n+m∑

r=1

∂ωr ⊗ er = −∑

p,q

(ωp ∧ ωq) ⊗ [ep, eq].

This equation amounts to stating that the ωp ∧ ωq component of 2∂er equals

ωr([ep, eq]) = 2dωr(ep, eq) = 2∂ωr(ep, eq),

which is correct.

Definition 1. Define Hk∂(g1,0) to be the kth cohomology ker ∂k/ Im∂k−1 of

the above complex; more precisely,

Hk∂(g1,0) =

ker(∂k : g∗(0,k) ⊗ g1,0 → g∗(0,k+1) ⊗ g1,0)∂k−1(g∗(0,k−1) ⊗ g1,0)

.

We shall interpret these spaces geometrically in the next subsection.

3.2. Dolbeault cohomology

Let Γ be a J-invariant co-compact lattice in G, and X = Γ\G the associatednilmanifold parameterizing left cosets. Let ψ : G → G/C be the quotient map,where C is the center of G. Since G is 2-step nilpotent, G/C is abelian. In terms ofthe abelian varieties F := C/C ∩Γ and M := ψ(G)/ψ(Γ), we obtain a holomorphicfibration

Ψ: X −→M

with fiber F .

Lemma 4. Let OX and ΘX be the structure sheaf and the tangent sheaf of X .For p � 1, the direct image sheaves with respect to Ψ are

RpΨ∗OX =∧p

c∗(0,1) ⊗OM = c∗(0,p) ⊗OM ,

RpΨ∗Ψ∗ΘM =∧p

c∗(0,1) ⊗ ΘM = c∗(0,p) ⊗ ΘM .


Proof. The second identity is a consequence of the first, and the projectionformula. To prove the first, note that, for any point m in M ,

(RpΨ∗OX)m = Hp(Ψ−1(m),OX) ∼= Hp(C,OX).

This has constant rank and, by Grauert’s Theorem, the direct image sheaf is locallyfree. As Ψ−1(m) is isomorphic to a complex torus, for all p � 1,

Hp(Ψ−1(m),OX) =∧pH1(Ψ−1(m),OX).

The vector bundle RpΨ∗OX is isomorphic to∧pR1Ψ∗OX . Since the space of vertical

(0,1)-forms is trivialized by the left-invariant (0,1)-forms given in Lemma 1, we have

RpΨ∗OX∼=

∧pR1Ψ∗OX

∼=∧p

c∗(0,1) ⊗OM ,

as required.

Lemma 5. Let OX and ΘX be the structure sheaf and the tangent sheaf of X .Then

Hk(X,OX) =∧k

g∗(0,1) = g∗(0,k),

Hk(X,Ψ∗ΘM ) =∧k

(g∗(0,1)) ⊗ t1,0 = g∗(0,k) ⊗ t1,0.

Proof. Consider the Leray spectral sequence with respect to the ∂-operator andthe holomorphic projection Ψ. One has

Ep,q2 = Hp(M,RqΨ∗OX), Ep,q

∞ ⇒ Hp+q(X,OX).

From the previous lemma, when q � 1,

Ep,q2 = Hp(M,

∧qc∗(0,1) ⊗OM ) =

∧qc∗(0,1) ⊗Hp(M,OM ) =

∧qc∗(0,1) ⊗

∧pt∗(0,1).

Note that every element in Ep,q2 is a linear combination of the tensor products

of vertical (0, q)-forms and (0, p)-forms lifted from the base. Since these forms areglobally defined and the differential d2 is generated by the ∂-operator, we haved2 = 0. It follows that the Leray spectral sequence degenerates at the E2-level.Therefore,

Hk(X,OX) =⊕

p+q=k

Ep,q2 =

∧k(c∗(0,1) ⊕ t∗(0,1)) =

∧kg∗(0,1).

Next, the spectral sequence for Ψ∗ΘM gives

Ep,q2 = Hp(M,RqΨ∗Ψ∗ΘM ), Ep,q

∞ ⇒ Hp+q(X,Ψ∗ΘM ).

Moreover, Ep,q2 is equal to

Hp(M,∧q

c∗(0,1) ⊗ ΘM ) =∧q

c∗(0,1) ⊗Hp(M,ΘM ) =∧q

c∗(0,1) ⊗∧p

t∗(0,1) ⊗ t1,0.

Elements in t1,0 are holomorphic vector fields on M and hence globally definedsections of Ψ∗ΘM on X . Elements in

∧qc∗(0,1) are pulled back to globally defined

(0, q)-forms on X . Crucially, elements in∧p

t∗(0,1) are globally defined holomorphic(0, p)-forms on X , and the operator d2 is identically zero. Therefore, the spectralsequence degenerates at E2. We have

Hk(X,Ψ∗ΘM ) =⊕

p+q=k

Ep,q2 =

∧k(c∗(0,1) ⊕ t∗(0,1)) ⊗ t1,0 =

∧k(g∗(0,1)) ⊗ t1,0,

as required.


Theorem 1. Let X be a 2-step nilmanifold with an abelian complex structure.There is a natural isomorphism Hk(X,ΘX) ∼= Hk

∂(g1,0).

Proof. On the manifold X , we have the exact sequence

0 → c1,0 ⊗OX → ΘX → Ψ∗ΘM → 0.

A piece of the corresponding long exact sequence is

· · · → c1,0 ⊗Hk(X,OX) → Hk(X,ΘX)

→ Hk(X,Ψ∗ΘM ) δk−→ c1,0 ⊗Hk+1(X,OX) → · · · .From the last section, the coboundary map is

δk : g∗(0,k) ⊗ t1,0 → g∗(0,k+1) ⊗ c1,0,

and so

Hk(X,ΘX) ∼= ker δk ⊕ g∗(0,k) ⊗ c1,0

δk−1(g∗(0,k−1) ⊗ t1,0).

We calculate the coboundary maps by chasing the following commutative dia-gram.

0 �� T ∗(0,k+1) ⊗ c1,0 �� T ∗(0,k+1) ⊗ ΘX�� T ∗(0,k+1) ⊗ Ψ∗ΘM

�� 0

0 �� T ∗(0,k) ⊗ c1,0 ��

��

T ∗(0,k) ⊗ ΘX��

��

T ∗(0,k) ⊗ Ψ∗ΘM��

��

0

The vertical maps are the ∂ for Dolbeault cohomology. More specifically, if ∇ is theChern connection, if ω is a (0, k)-form and if V is a vector field of type (1, 0), then

∂(ω ⊗ V ) = ∂ω ⊗ V + (−1)kω ∧ ∂∇V.In the following computation, we let {ep} be a left-invariant basis for g1,0 and {ωp}the dual basis.

Let ω be a (0, k)-form. Let V be an element in t1,0, considered as a holomorphicvector field on M and a holomorphic section of Ψ∗ΘM . Let V be a smooth liftingof this section to a section of ΘX . Then

δk(ω ⊗ V ) = ∂ω ⊗ V + (−1)k∑

p

ω ∧ ωp ⊗ [ep, V ]1,0.

The element Tj in gC could be considered as a holomorphic vector field on M .It could also be considered as a smooth vector field on X . Considering the latter tobe a lifting of the former and applying the above formula, we see that δk = ∂k ong∗(0,k) ⊗ t1,0. Now ∂k(g∗(0,k) ⊗ c1,0) = 0, and so

δk−1(g∗(0,k−1) ⊗ t1,0) = ∂k−1(g∗(0,k−1) ⊗ g1,0).

Since the Lie algebra g is 2-step nilpotent, Im ∂k−1 ⊆ g∗(0,k) ⊗ c1,0. Also, we have

ker δk = ker ∂k ∩ (g∗(0,k) ⊗ t1,0).


Therefore,

Hk(X,ΘX) = ker ∂k ∩ (g∗(0,k) ⊗ t1,0) ⊕ g∗(0,k) ⊗ c1,0

Im ∂k−1

=ker ∂k ∩ (g∗(0,k) ⊗ t1,0 ⊕ g∗(0,k) ⊗ c1,0)

Im ∂k−1

= Hk∂(g), (13)

as stated.

To summarize the results in this section, we shall say that a tensor on X isinvariant if its pull-back to G by the quotient map is invariant by left-translationby G. Lemma 5 and Theorem 1 then allow us to formulate the following theorem.

Theorem 2. The Dolbeault cohomology on X with coefficients in the structureand tangent sheaf can be computed using invariant forms and invariant vectors.

Although the above proof relies on the 2-step property, one might expect thatthis result has a more general validity, at least in the nilpotent context. For anindependent approach to this problem, see [7].

4. Deformation theory

We shall shortly be in a position to apply the Kuranishi method [16] to con-struct deformations. However, first we shall exhibit harmonic representatives in theDolbeault cohomology groups.

4.1. Harmonic theory

Theorem 1 reduces the question to finite-dimensional vector spaces, and we maychoose an invariant Hermitian structure onX of the type mentioned after Lemma 1.We use the resulting inner product on g∗(0,k) ⊗ g1,0 to define the orthogonal com-plement of Im ∂k−1 in ker ∂k. Denote this space by Im⊥ ∂k−1.

Theorem 3. The space Im⊥ ∂k−1 is a space of harmonic representatives forthe Dolbeault cohomology Hk(X,ΘX) on the compact complex manifold X .

Proof. It suffices to prove that an element∑

p

σp ⊗ ep ∈ Im⊥ ∂k−1 ⊆ g∗(0,k) ⊗ g1,0 (14)

is ∂∗-closed on the manifold X .

Any section of the trivial bundle over X with fiber g∗(0,k−1) ⊗ g1,0 is a sum ofelements of the type fη ⊗ V , where f is a smooth function, η ∈ g∗(0,k−1) andV ∈ g(1,0). By Lemma 1, σp and η are ∂-closed. Using double angular brackets forthe L2 inner product and summing over repeated indices, we calculate

〈〈∂∗(σp ⊗ ep), fη ⊗ V 〉〉 = 〈〈σp, ∂(fη)〉〉〈ep, V 〉 + (−1)k−1〈〈σp ⊗ ep, fη ∧ ∂V 〉〉= 〈〈∂∗

σp, fη〉〉〈ep, V 〉 + (−1)k−1〈〈σp ⊗ ep, fη ∧ ∂V 〉〉.The basis {ωi, ωα} of Lemma 1 determines a complex volume form that we may

use to identify ∂∗

with ± ∗∂ ∗, where ∗ is the corresponding SU(n + m) invariant


antilinear mapping g∗(0,k) → g∗(0,n+m−k). It follows that ∂∗σ = 0. The remaining

term

〈〈σp ⊗ ep, fη ∧ ∂V 〉〉 =�X

f〈σp ⊗ ep, η ∧ ∂V 〉 = 〈σp ⊗ ep, ∂(η ⊗ V )〉�X

f

vanishes by assumption (14).

Corollary 1. Let µ ∈ g∗(0,k) ⊗ g1,0. Then ∂∗µ with respect to the L2 norm

on the compact manifold X is equal to ∂∗µ with respect to the Hermitian inner

product on the finite-dimensional vector spaces g∗(0,k) ⊗ g1,0.

Proof. This follows from the displayed formulae in the previous proof.

4.2. The Schouten–Nijenhuis bracket

If ω ⊗ V and ω′ ⊗ V ′ are vector-valued (0,1)-forms representing elements inH1(X,ΘX), their product with respect to the Schouten–Nijenhuis bracket is avector-valued (0,2)-form

{·, ·} : H1(X,ΘX) ×H1(X,ΘX) → H2(X,ΘX).

It is defined at the level of forms by

{ω ⊗ V, ω′ ⊗ V ′} = ω′ ∧ LV ′ω ⊗ V + ω ∧ LV ω′ ⊗ V ′ + ω ∧ ω′ ⊗ [V, V ′].

Via the isomorphism with Lie algebra cohomology, elements in H1(X,ΘX) lie inIm⊥ ∂0. Since the vector and form parts are all left-invariant, ιV ω′ is a constant.Therefore, LV ω

′ = dιV ω′ + ιV dω

′ = ιV dω′, and

{ω ⊗ V, ω′ ⊗ V ′} = ω′ ∧ ιV ′dω ⊗ V + ω ∧ ιV dω′ ⊗ V ′ + ω ∧ ω′ ⊗ [V, V ′].

The complex structure is abelian, so [V, V ′] = 0 for all (1, 0)-vectors, and

{ω ⊗ V, ω′ ⊗ V ′} = ω′ ∧ ιV ′dω ⊗ V + ω ∧ ιV dω′ ⊗ V ′. (15)

Using the vector space direct sum g = t ⊕ c, we write

g∗(0,1) ⊗g1,0 = (t∗(0,1) ⊗ t1,0)⊕ (c∗(0,1) ⊗ t1,0)⊕ (c∗(0,1) ⊗ c1,0)⊕ (t∗(0,1) ⊗ c1,0). (16)

If ω⊗V ∈ t∗(0,1)⊗c1,0 then dω = 0, because all elements in t∗(k,l) are closed. On theother hand, dω′ ∈ t∗(1,1). Since ιV dω′ = 0 for V ∈ c1,0, we have

{t∗(0,1) ⊗ c1,0, g∗(0,1) ⊗ g1,0} = 0. (17)

In order to compute {µ,ν} on Im⊥ ∂0, we compute the bracket amongst elementsin the obvious basis. In view of (17), we need to calculate the brackets arisingfrom the first three summands in (16). There are six types of bracket to calculate.Since ωk and ωj are closed,

{ωj ⊗ Ti, ωk ⊗ Tl} = 0,

{ωj ⊗ Ti, ωα ⊗ Tl} = ωj ∧ ιTidω

α ⊗ Tl = −Eα

ikωj ∧ ωk ⊗ Tl,

{ωj ⊗ Ti, ωα ⊗Wσ} = ωj ∧ ιTidω

α ⊗Wσ = −Eα

ihωj ∧ ωh ⊗Wσ,

{ωα ⊗ Tl, ωβ ⊗ Tj} = −Eβ

lhωα ∧ ωh ⊗ Tj − E

α

jhωβ ∧ ωh ⊗ Tl,

{ωα ⊗ Tl, ωβ ⊗Wγ} = ωα ∧ ιTl

dωβ ⊗Wγ = −Eβ

lhωα ∧ ωh ⊗Wγ ,

{ωα ⊗Wβ , ωγ ⊗Wδ} = 0.

(18)


The above formulae allow us to calculate {µ,ν}. If µ is given in coordinates asin (12) and ν similarly, then, suppressing summation signs, we have

{µ,ν} = −(µijν

�α + νi

jµ�α)E

α

ik ωj ∧ ωk ⊗ T�

− (µ�αν

jβ + ν�

αµjβ)(E

β

�kωα ∧ ωk ⊗ Tj + E

α

jkωβ ∧ ωk ⊗ T�)

− (µijν

δγ + νi

jµδγ)E

γ

ik ωj ∧ ωk ⊗Wδ

− (µiαν

δγ + νi

αµδγ)E

γ

ik ωα ∧ ωk ⊗Wδ. (19)

In particular,

{µ,µ} = −2µijµ

�αE

α

ik ωj ∧ ωk ⊗ T�

− 2µ�αµ

jβ(E

β

�kωα ∧ ωk ⊗ Tj + E

α

jkωβ ∧ ωk ⊗ T�)

− 2µijµ

δγE

γ

ik ωj ∧ ωk ⊗Wδ − 2µi

αµδγE

γ

ik ωα ∧ ωk ⊗Wδ, (20)

which is of course an element of g∗(0,2) ⊗ g1,0.

4.3. Kuranishi theory

To construct deformations, we apply Kuranishi’s recursive formula. Let {β1,. . . , βN} be an orthonormal basis of the harmonic representatives of H1(X,ΘX).For any vector t = (t1, . . . , tN ) in CN , let

µ(t) = t1β1 + . . .+ tNβN . (21)

We set φ1 = µ, and next define φr inductively for r � 2.Consider the ∂-operator on X with respect to the Hermitian metric h previously

defined, its adjoint operator ∂∗

and the Laplacian

� = ∂∂∗

+ ∂∗∂. (22)

Let G be the corresponding Green’s operator that inverts � on the orthogonalcomplement of the space of harmonic forms, and let {·, ·} denote the Schouten–Nijenhuis bracket. Then we set

φr(t) =12

r−1∑

s=1

∂∗G{φs(t),φr−s(t)} =

12

r−1∑

s=1

G∂∗{φs(t),φr−s(t)}, (23)

and consider the formal sum

Φ(t) =∑

r�1

φr. (24)

Let {γ1, . . . , γM} be an orthonormal basis for the space of harmonic (0, 2)-formswith values in ΘX . Define fk(t) to be the L2 inner product 〈〈{Φ(t),Φ(t)}, γk〉〉.Kuranishi theory asserts the existence of ε > 0 such that

{t ∈ CN : |t| < ε, f1(t) = 0, . . . , fM (t) = 0} (25)

forms a locally complete family of deformations of X . We shall denote this set byKur. For each t ∈ Kur, the associated sum Φ = Φ(t) satisfies the integrabilitycondition

∂Φ + 12{Φ,Φ} = 0 (26)

that now follows from (23) and the definition of G.


More explicitly, we may treat Φ is a linear map from (0, 1)-vectors to (1, 0)-vectors. It determines a complex structure on our manifold X whose distributionof (0,1)-vectors is given by

Sj = T j + Φ(T j),

Vα = Wα + Φ(Wα).(27)

This set of equations is analogous to the gauge-theoretic definition of a connectionas dA = d + A, where A is a matrix of 1-forms. In principal bundle language, dA

determines a horizontal distribution formed from the flat one by adding A as avertical component. Then (26) is the analogue of setting the curvature of dA to bezero, and assures us that the new distribution (27) is closed under Lie bracket.

We are now ready to make precise the first theorem of the Introduction, namelythe following.

Theorem 4. Let G be a 2-step nilpotent Lie group with co-compact subgroupΓ, and let J be an abelian invariant complex structure on X = Γ\G. Then thedeformations arising from J parameterized by (25) are all invariant complex struc-tures.

Proof. It suffices to show that every term in the power series (24) lies in g∗(0,1)⊗g1,0. We shall prove this by induction. By Theorem 2, φ1 = µ belongs to this space.

Assume that φs ∈ g∗(0,1) ⊗ g1,0 for all 1 � s � r − 1. The computations of § 4.2show that {φs,φr−s} is always contained in g∗(0,2) ⊗ g1,0. This space decomposesinto a direct sum of the harmonic part H and its orthogonal complement H⊥. Letπ0 denote projection to the subspace H⊥.

As a linear map on a finite-dimensional space, � is an isomorphism on theorthogonal complement of its kernel. In other words, it maps H⊥ isomorphicallyonto itself. As the inverse operator of the Laplace operator, the Green’s operator isan isomorphism from H⊥ to itself, and sends harmonic elements to zero. In otherwords,

G{φs,φr−s} = Gπ0{φs,φr−s} ⊆ H⊥.

In particular, it is an invariant tensor. Corollary 1 shows that ∂∗G{φs,φr−s} is

again an invariant tensor. The same is true of φr. By induction, (24) is an infiniteseries of invariant tensors.

5. Deformations leading to abelian structures

In the light of Theorem 4, we are now ready to identify deformations of J leadingnot just to invariant complex structures, but to abelian ones.

Given an element µ = µ(t) in the virtual parameter space H1(X,ΘX) as in (21),we apply the preceding method to generate the infinite series (24), and consider(27). For the latter to define an abelian complex structure, the Lie bracket of anypair of (0, 1)-vectors must in fact vanish identically. In this case, we shall say thatµ generates an abelian deformation. Such an assumption leads to the followingequations:

[Sj , Sk] = 0, 1 � j, k � n; (28)


[Sj , Vα] = 0, 1 � j � n, n+ 1 � α � n+m; (29)

[Vα, Vβ] = 0, n+ 1 � α, β � n+m. (30)

Since Wα is in the center,

[Vα, Vβ] = [Φ(Wα),Φ(Wβ)],

and this vanishes since the original complex structure is abelian. Therefore, equation(30) is satisfied automatically.

Let us examine the infinitesimal consequence of the first two equations. Let trepresent a real variable, and replace µ by tµ so that Φ becomes

∑trφr. Then in

the notation of (12), equation (28) leads to

0 =d

dt

∣∣∣∣t=0

[Sj , Sk] = [T j ,φ1T k] + [φ1T j, T k] = [T j , µikTi] + [µi

jTi, T k]

= (µikE

αji − µi

jEαki)Wα + (µi

kFαji − µi

jFαki)Wα. (31)

The coefficient of Wα vanishes when ∂µ = 0, by Lemma 2. Equation (29) leads to

0 =d

dt

∣∣∣∣t=0

[Sj , Vα] = [T j ,φ1Wα] = [T j , µiαTi]

= µiαE

βjiWβ + µi

αFβjiWβ. (32)

The coefficient of Wβ is again 0 when µ is ∂-closed.The above calculations give a set of necessary conditions limiting the type of

deformations that one needs to consider. They motivate the following definition.

Definition 2. A form µ given in coordinates as in (12) is said to satisfyCondition A if

∑

i

(µijF

αki − µi

kFαji) = 0 and

∑

i

µiαF

βji = 0,

for each j, k, α, β.

It is striking that these conditions are completely analogous to those of Lemma 2.In view of (31) and (32), we can now state the following proposition.

Proposition 1. A parameter µ represents an infinitesimal abelian deformationif and only if it is ∂-closed and satisfies Condition A.

Next suppose that µ and ν are vector-valued 1-forms that are both ∂-closedand satisfy Condition A. Since E

α

ij = −Fαji, every term in (19) is equal to 0. For

example, the first term −µijν

�αE

α

ikωj ∧ ωk ⊗ T� is equal to

µijν

�αF

αkiω

j ∧ ωk ⊗ T� = ν�α(µi

jFαki − µi

kFαji)ω

j ⊗ ωk ⊗ T� = 0,

and similarly every term in {µ,ν} is equal to zero. In particular, {µ,µ} = 0.Using the recursive formula (23), the higher order terms are all equal to zero, and

so the series Φ and µ coincide by construction. Furthermore, {Φ,Φ} = {µ,µ} =0, and there is no additional obstruction to integrability. Therefore, we have thefollowing proposition.


Proposition 2. On a 2-step nilmanifold X with abelian complex structure,an element in H1(X,ΘX) is infinitesimally abelian only if it is integrable to a1-parameter family of abelian complex structures.

Our main result concerning the deformation of abelian complex structures is asfollows.

Theorem 5. On a 2-step nilmanifold with abelian complex structure, a para-meter µ in g∗(0,1)⊗g1,0 generates an abelian deformation if and only if it is ∂-closedand satisfies Condition A.

Proof. If µ generates an abelian deformation, it is infinitesimally abelian. ByProposition 1, the form is ∂-closed and satisfies Condition A.

Conversely, if Φ is ∂-closed, it represents a cohomology class in H1(X,Θ). Sinceit also satisfies Condition A, it is infinitesimally abelian. By Proposition 2, itrepresents an integrable abelian complex structure.

5.1. Fully abelian deformations

We are curious to know when the entire virtual parameter space H1(X,ΘX)integrates to abelian complex structures.

Theorem 6. Let X = Γ\G be a compact 2-step nilmanifold endowed with anabelian complex structure. Suppose that every direction of the virtual parameterspace is integrable to a 1-parameter family of abelian complex structures and thatthe real dimension of the center of Lie algebra g is equal to 2. Then g is isomorphicto the direct sum of a Heisenberg algebra and a 1-dimensional abelian algebra.

Proof. We may as well drop the index α in Eαij since it only takes on one value

given the hypothesis on the center (see (2)). This feature makes the subsequentconstruction possible.

Given the structural constants, for each set of j, k, l,m, choose an element µ ing∗(0,1)⊗g1,0 by setting µl

k = Ekm and µmj = Ejl and setting all other terms to zero.

By Lemma 2, each such µ is closed. By equation (20), such µ satisfies the equation{µ,µ} = 0 and therefore there is no obstruction for it to represent an integrablecomplex structure.

By hypothesis, µ represents an abelian complex structure. By Condition A,

EkmElj − EjlEmk = 0.

It follows that |Ekm|2 = |Emk|2 for all k and m.If every Ekm vanishes then the algebra is abelian. On the other hand, if at least

one Ekm is non-zero, then Emk �= 0. For every Ejl �= 0, the ratio

Ejl

Elj=Ekm

Emk

is independent of the choice of j, l. Hence, there exists a real number θ such that

eiθEjl = Elj , (33)


for every pair of indices (j, l). It follows that

[T j , Tl] = EjlW + FjlW = EjlW − EljW = EjlW − eiθEjlW.

ChoosingDjl = ei(π+θ)/2Ejl, U = e−i(π+θ)/2W

gives[T j , Tl] = Djl(U + U).

With (33), we find that the matrix (Djl) is skew-Hermitian. If we now choosea basis of (0, 1)-vectors so that the matrix D is diagonal, the diagonal entries arepurely imaginary or zero. The restriction on the central dimension forces the matrixD to be a constant multiple of the identity matrix. It follows that the structuralequations exactly mirror those of the Heisenberg algebra as seen in (5).

Example 6. There exist examples satisfying the first hypothesis of the theorem,but not the second. To see this, take g to be the real 8-dimensional Lie algebra withnon-zero complex structural equations

[T 1, T1] = W3 +W 3, [T 2, T2] = W4 +W 4,

and real 4-dimensional center. By Lemma 2, µ21 = µ1

2 = 0, and µhα = 0 for 1 � h � 2

and 3 � α � 4. It follows that

H1(X,ΘX) = 〈ω1 ⊗ T1, ω2 ⊗ T2, ω

1 ⊗W4, ω2 ⊗W3〉,

and one may check that each direction is integrable to abelian complex structures.Globally, the associated compact complex manifold is the product of two primaryKodaira surfaces.

6. Six-dimensional structures

In dimension 6, there are precisely six classes of 2-step groups or nilmanifoldswith an abelian complex structure [21]. Namely, the abelian group R6, the productH5 × R1 of a 5-dimensional Heisenberg group with a 1-dimensional group, theproductH3×R3 of the 3-dimensional Heisenberg group with a 3-dimensional abeliangroup, the product H3×H3 of two 3-dimensional Heisenberg groups, and the groupsP6 and W6. These were encountered in § 2.1.

We shall use Example 4 to illustrate that results in this article produce adequateinformation for finding the parameters for integrable and abelian deformations.Using (9), consider the basis

T1 = X1 − iX2, T2 = X3 + iX4, W = Z5 + iZ6

of g1,0; the corresponding basis for g∗(0,1) is {ω1, ω2, ω}. The structural equationsyield

[T 1, T2] = −W,so that

E12 = −1, F21 = 1,

and all other structural constants are equal to zero. In particular,

dω = ω1 ∧ ω2, ιT1dω = ω2, ιT2dω = 0. (34)


Mimicking the proof of Lemma 2, any element µ ∈ g∗(0,1) ⊗ g1,0 can be writtenas

Φ = µijω

j ⊗ Ti + µi3ω ⊗ Ti + µ3

jωj ⊗W + µ3

3ω ⊗W,

and

∂µ = µ22ω

2 ∧ ω1 ⊗W + µ23ω ∧ ω1 ⊗W.

This shows that µ is closed if and only if µ22 = µ2

3 = 0. Since ∂T2 = −ω1 ⊗W , thespace of harmonic elements is in the orthogonal complement of ω1 ⊗W . Therefore,

dimH1(X,ΘX) = dim{µ22 = µ2

3 = µ31 = 0} = 6.

Using Definition 2, we see immediately that µ satisfies Condition A if and only if

µ11 = µ1

3 = 0. (35)

The number of parameters corresponding to abelian deformations is therefore 4.Alternatively, we can count the number of integrable parameters, disregarding

the abelian issue. To do so, we employ the recursive formula from § 4.3, and firstcalculate the self-bracket of a harmonic representative µ. Using (34), (15), and (18),we deduce that

{µ,µ} = {µ11ω

1 ⊗ T1 + µ13ω ⊗ T1 + µ3

3ω ⊗W, µ11ω

1 ⊗ T1 + µ13ω ⊗ T1 + µ3

3ω ⊗W}= µ1

3(2µ11ω

1 ∧ ω2 ⊗ T1 + µ13ω ∧ ω2 ⊗ T1 + 2µ3

3ω ∧ ω2 ⊗W )

− 2µ11µ

33∂(ω2 ⊗ T2).

Using (23), we take

φ2 = µ11µ

33ω

2 ⊗ T2.

This quadratic correction term exactly corresponds to the equation d = −av in [15,Proposition 4.2].

If we set Φ = µ + φ2, then (26) becomes

µ13(2µ

11ω

1 ∧ ω2 ⊗ T1 + µ13ω ∧ ω2 ⊗ T1 + 2µ3

3ω ∧ ω2 ⊗W ) = 0.

The resulting deformation is therefore integrable if and only if µ13 = 0, so there is a

total of five integrable parameters. As predicted by Theorem 5, the obstruction µ13

already features in the abelian equations (35).Let g denote a real 6-dimensional nilpotent Lie algebra admitting a complex

structure. The table in [21, Appendix] displays, for each such g, the complexdimension of the space C(g) of invariant complex structures at a smooth pointof one of its connected components. This was done with little regard for whencomplex structures are equivalent, in the knowledge that subsequent work wouldclarify the findings. Table 1 compares these computations with results yielded bythe techniques of this paper. The last five columns display the complex dimension

(i) d of C(g),(ii) h0 of the space dimH0(X,ΘX) of infinitesimal automorphisms,(iii) h1 of the virtual parameter space H1(X,ΘX),(iv) of the space Kur of (25), or the number of integrable parameters,(v) of the subspace Abel of Kur describing abelian deformations,

relative to the complex structures defined by (4), (6), (7), (9), and (10), for each ofthe last five rows in turn.


If J is an invariant complex structure with unobstructed deformations on anilmanifold, C(g) has the same dimension as the kernel of

∂ : g∗(0,1) ⊗ g1,0 → g∗(0,2) ⊗ g1,0,

whereas dim Kur = h1. Since the dimension of the image ∂(g∗(0,0) ⊗ g1,0) equals3 − h0, we deduce further that

d = 3 − h0 + h1

if J is a generic point of C(g).At points of C(g) where h0 jumps to a higher value, the Kuranishi method is

unable to detect the additional equivalences that come into play at neighboringpoints where the symmetry group drops. Consequently, we can only assert thatd + h0 − 3 is an upper bound for dim Kur. In Table 1, these two numbers onlydisagree for W6, and this is because J1 was ‘too’ special a choice at which to carryout the computations. If we work instead at a nearby point J ′ corresponding toµ3

3 �= 0, then h0 = 1 and the dimensions of Kur and Abel drop to 4 and 3. Thisis because the orbit J ′ ·W6 under right-translation by the group has dimension 2,whereas dim(J ·W6) = 1.

A more extreme example, not tabulated, is that of the non-abelian complexstructure J0 on Γ\W6 for which h0 = 3, h1 = d = dim Kur = 6 and dim Abel = 0[19, 21].

6.1. Final examples

In general, information on H1 and abelian deformations can be extracted alge-braically using Lemma 2 and Definition 2. The computation of Kur is more chal-lenging, though it is useful to realize that every parameter is integrable whenh1 = dim Abel.

In our last two examples, the first applies the theory of § 4.3, whereas the secondrelies on this theory to pass directly to a calculation with invariant differentialforms.

Example 7. For (Γ\H3) × (Γ\H3) (see Example 3), we may take

T1 = 12 (X1 − iY1), T2 = 1

2 (X2 − iY2), W = 12 (Z1 − iZ2).

The associated complex structural equations are

[T 1, T1] = − 12 i(W +W ), [T 2, T2] = 1

2 (W −W ).

Table 1. Deformation parameters.

d h0 h1 dim Kur dimAbel

T 6 9 3 9 9 9(Γ\H5) × S1 6 1 4 4 4(Γ\H3) × T 3 7 2 6 6 6

(Γ\H3) × (Γ\H3) 6 1 4 4 3Γ\W6 6 2 6 5 4Γ\P6 6 1 4 4 3


In terms of the dual basis {ωi}, any harmonic representative of H1 is a linearcombination of

ω1 ⊗ T1, ω2 ⊗ T2, ω ⊗W, ω1 ⊗ T2 + iω2 ⊗ T1.

Ifµ = µ1

1ω1 ⊗ T1 + µ2

2ω2 ⊗ T2 + µ3

3ω ⊗W + µ21(ω

1 ⊗ T2 + iω2 ⊗ T1),

we obtainΦ = µ − µ3

3µ21(ω

1 ⊗ T2 − iω2 ⊗ T1).

Then (27) defines an integrable complex structure.

Example 8. The complex structural equations corresponding to Example 5can be written in the form dω1 = 0 = dω2 and

2dω3 = iω1 ∧ ω1 + ω1 ∧ ω2 − ω1 ∧ ω2 = iω11 + ω12 − ω12,

in which the last expression is an abbreviation of the middle one. By [15, Theo-rem 1.1], any invariant complex structure J ′ sufficiently near to J has a basis of(1, 0)-forms that can be written

α1 = ω1 + Φ11ω

1 + Φ12ω

2,

α2 = ω2 + Φ21ω

1 + Φ22ω

2,

α3 = ω3 + Φ31ω

1 + Φ32ω

2 + Φ33ω

3.

(36)

This is a dual version of (27), and the integrability condition (26) amounts to theassertion that (dα3)0,2 = 0, or equivalently

0 = 2dα3 ∧ α1 ∧ α2

= [(iω11 + ω12 − ω12) + Φ33(iω

11 − ω12 + ω12)] ∧ [−Φ12ω

22 + Φ22ω

12 + Φ11ω

12]

= [−iΦ12(1 + Φ3

3) + (1 − Φ33)(Φ

11 − Φ2

2)]ω1122.

Thus, (36) defines an integrable complex structure on condition that

i(1 + Φ33)Φ

12 = (1 − Φ3

3)(Φ11 − Φ2

2),

and the coefficients in (36) are sufficiently small (in particular, |Φ33| < 1).

In this case, the Kuranishi series (24) is infinite, as it is not possible to express onecoefficient as a polynomial in the others. The term Φ3

1ω1 + Φ3

2ω2 can be reduced

to zero by a suitable right-translation of J , and therefore plays no role in theequivalence problem. It follows that dim Kur = 4. The abelian condition

dα3 ∧ α1 ∧ α2 = 0

can be worked out in the same way, and forces Φ12 = 0 and Φ1

1 = Φ22, so dim

Abel = 3.

Table 1 and the examples allow us to infer the following.(i) It is possible that every direction in the virtual parameter space is integrable

but only some are tangent to abelian deformations. This occurs for (Γ\H3)×(Γ\H3)and Γ\P6.

(ii) It is also possible that some directions are obstructed, irrespective of theabelian condition. An example is Γ\W6. This phenomenon was described in [21,Lemma 4.3], and contrasts with the unobstructed deformation theory for (Γ\W6, J0).


(iii) The centers of T 6 and (Γ\H3)×T 3 certainly have dimension greater than 1,and these examples do not therefore contradict Theorem 6.All these observations demonstrate the subtle dependence of dim Kur and dim Abelon the underlying algebraic structure of the group G.

The techniques of this paper can in theory be applied to study deformations ofthe compact quotients of the six series of 2-step nilmanifolds with abelian complexstructures in any complex dimension.

Other work of the authors shows that in many cases an explicit description ofKur, and indeed a global moduli space, is possible [13, 17]. It is also realistic toseek to describe the quotient of the space C(g) by the group of Aut(g) of Lie algebraautomorphisms of g, at least near a generic point of C(g). In this case, in the W6

example, C(g)/Aut(g) is locally isomorphic to the quotient of Kur by the group ofouter automorphisms of g [11, § 5].

Acknowledgements. Some results herein were reported in [17] to fulfill a degreerequirement of C. Maclaughlin.

References

1. E. Abbena, S. Garbiero and S. Salamon, ‘Almost Hermitian geometry on six-dimensionalnilmanifolds’, Ann. Sc. Norm. Super. Pisa 30 (2001) 147–170.

2. M. L. Barberis, ‘Hypercomplex structures on four-dimensional Lie groups’, Proc. Amer.Math. Soc. 125 (1997) 1043–1054.

3. M. L. Barberis, ‘Abelian hypercomplex structures on central extensions of H-type Liealgebras’, J. Pure Appl. Algebra 158 (2001) 15–23.

4. M. L. Barberis and I. Dotti, ‘Abelian complex structures on solvable Lie algebras’, J. LieTheory 14 (2004) 25–34.

5. M. L. Barberis, I. G. Dotti Miatello and R. J. Miatello, ‘On certain locally homo-geneous Clifford manifolds’, Ann. Global Anal. Geom. 13 (1995) 513–518.

6. C. Birkenhake and H. Lange, Complex Tori, Progress in Mathematics 177 (Birkhauser,Boston, 1999).

7. S. Console and A. Fino, ‘Dolbeault cohomology of compact nilmanifolds’, Transform.Groups 6 (2001) 111–124.

8. L. A. Cordero, M. Fernandez, A. Gray and L. Ugarte, ‘Compact nilmanifolds withnilpotent complex structures: Dolbeault cohomology’, Trans. Amer. Math. Soc. 352 (2000)5405–5433.

9. I. G. Dotti and A. Fino, ‘Abelian hypercomplex 8-dimensional nilmanifolds’, Ann. GlobalAnal. Geom. 18 (2000) 47–59.

10. I. G. Dotti and A. Fino, ‘HyperKahler torsion structures invariant by nilpotent Lie groups’,Classical Quantum Gravity 19 (2002) 551–562.

11. A. Fino, M. Parton and S. Salamon, ‘Families of strong KT structures in six dimensions’,Comment. Math. Helv. 79 (2004) 317–340.

12. P. Gauduchon, ‘Hermitian connections and Dirac operators’, Boll. Unione Mat. Ital. 11-B(1997) 257–288.

13. G. Grantcharov, C. Maclaughlin, H. Pedersen and Y. S. Poon, ‘Deformations ofKodaira manifolds’, Glasgow Math. J. 46 (2004) 259–281.

14. G. Grantcharov and Y. S. Poon, ‘Geometry of hyper-Kahler connections with torsion’,Commun. Math. Phys. 213 (2000) 19–37.

15. G. Ketsetzis and S. Salamon, ‘Complex structures on the Iwasawa manifold’, Adv. Geom.4 (2004) 165–179.

16. M. Kuranishi, ‘On the locally complete families of complex analytic structures’, Ann. ofMath. (2) 75 (1962) 536–577.

17. C. Maclaughlin, ‘Integrable and abelian deformations of abelian complex structures on2-step nilmanifolds, PhD Thesis, University of California, Riverside, 2003.

18. I. A. Malcev, ‘On a class of homogeneous spaces’, Amer. Math. Soc. Transl. 9 (1962)276–307.

19. I. Nakamura, ‘Complex parallelizable manifolds and their small deformations’, J. DifferentialGeom. 10 (1975) 85–112.


20. K. Nomizu, ‘On the cohomology of compact homogeneous spaces of nilpotent Lie groups’,Ann. of Math. (2) 59 (1954) 531–538.

21. S. Salamon, ‘Complex structures on nilpotent Lie algebras’, J. Pure Appl. Algebra 157 (2001)311–333.

C. Maclaughlin and Y. S. PoonDepartment of MathematicsUniversity of California at RiversideRiversideCA 92521USA

[email protected]@math.ucr.edu

H. PedersenDepartment of Mathematics and

Computer ScienceUniversity of Southern DenmarkCampusvej 55Odense MDK–5230Denmark

[email protected]

S. SalamonDipartimento di MatematicaPolitecnico di TorinoCorso Duca degli Abruzzi 2410129 TorinoItaly

[email protected]

DEFORMATION OF 2-STEP NILMANIFOLDS WITH ABELIAN … · a(2n + 1)-dimensional Heisenberg group, then...

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